format-ladder
At its defaults it draws refinement thresholds by format, against a problem at κ = 1000. A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
format-ladder is one function in lib/figures/mixed.js —
mixed precision — refinement, and the threshold at 1/u. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
logKappa: 4
The arguments are the ones Buying the accuracy back passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
logKappa: 3
The arguments are the ones The other half of a format passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A horizontal bar chart of the condition number at which iterative refinement stops working, one bar per floating-point format, with a marked line at the condition number of the problem.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
16 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
and κ·u is κ times it for fp16 — asserted 3 times
fp16's threshold is exactly 1/u — asserted 3 times
and κ·u is κ times it for bfloat16
and κ·u is κ times it for tf32
at least one format clears this κ
bfloat16's threshold is exactly 1/u
fp16 and tf32 share a threshold
fp16 has at least bfloat16's threshold
fp32 has at least tf32's threshold
fp64 has at least fp32's threshold
tf32 has at least fp16's threshold
tf32's threshold is exactly 1/u
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Buying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.
The arithmetic underneathThe other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.
The arithmetic underneathWhere the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.